We show that for every $n$-point metric space $M$ and positiveinteger $k$, there exists a spanning tree $T$ with unweighteddiameter $O(k)$ and weight $w(T) = O(k cdot n^{1/k}) cdotw(MST(M))$, and a spanning tree $T'$ with weight $w(T') = O(k) cdotw(MST(M))$ and unweighted diameter $O(k cdot n^{1/k})$. Moreover,there is a designated point $rt$ such that for every other point$v$, both $dist_T(rt,v)$ and $dist_{T'}(rt,v)$ are at most$(1+epsilon)cdot dist_M(rt,v)$, for an arbitrarily small constant$epsilon ≫ 0$.We prove that the above tradeoffs are emph{tight up to constantfactors} in the entire range of parameters. Furthermore, our lowerbounds apply to a basic one-dimensional Euclidean space. Finally,our lower bounds for the particular case of unweighted diameter$O(log n)$ settle a long-standing open problem in ComputationalGeometry.
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机译：我们表明，对于每一种$ n $ -point度量空间$ m $和partioninteger $ k $，有一个生成的树$ t $ t $ t $ with unwighteddiameter $ o（k）$和wey $ w（t）= o（k cdot n ^ {1 / k}）cdotw（mst（m））$，以及一个生成的树$ t'$ with $ w（t'）= o（k）cdotw（MST（m））$和未加权直径$ o （k cdot n ^ {1 / k}）$。此外，有一个指定的点$ RT $，使其每隔一点$ V $，$ dist_t（rt，v）$和$ dist_ {t'}（RT，V）$至多$（1 + epsilon ）CDOT DIST_M（RT，V）$，用于任意小的常数$ epsilon»0 $。我们证明了上述权衡在整个参数范围内都是表情符号{紧固到CONSTALICTORS。此外，我们的下游适用于基本的一维欧几里德空间。最后，我们的下限对于特定的直径US（log n）$稳定在计算结果中的长期开放问题。
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